Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
41
What is the equation for the axis of symmetry of the parabola y = ax^2 + bx + c?
Answer:
x = -b / 2a
The axis of symmetry is the vertical line that divides the parabola into two mirror images. It passes through the vertex, and its equation is universally x = -b / (2a).
42
Find the minimum value of the quadratic function y = x^2 - 6x + 10.
Answer:
1
Because a = 1 (a > 0), the function has a minimum at the vertex. The x-coordinate is x = -(-6) / 2(1) = 3. Substituting x = 3 back into the equation gives y = (3)^2 - 6(3) + 10 = 9 - 18 + 10 = 1. The minimum value is 1.
43
What is the maximum value of the function y = -x^2 + 4x?
Answer:
4
Because a = -1 (a < 0), the function has a maximum value at its vertex. The x-coordinate is x = -b / 2a = -4 / -2 = 2. Plugging x = 2 into the function gives y = -(2)^2 + 4(2) = -4 + 8 = 4. The maximum value is 4.
44
Does the parabola defined by y = -x^2 + 4x - 1 open upwards or downwards?
Answer:
Downwards
The direction a parabola opens is determined by the coefficient of the x^2 term (a). If a < 0, the parabola opens downwards. Since a = -1 here, the parabola opens downwards.
45
Find the y-intercept of the quadratic function y = 2x^2 + 3x - 5.
Answer:
-5
The y-intercept of any function occurs where x = 0. Substituting x = 0 into the function y = 2(0)^2 + 3(0) - 5 leaves y = -5. The y-intercept is -5.
46
What is the x-coordinate of the vertex of the parabola given by y = x^2 - 4x + 3?
Answer:
2
The x-coordinate of the vertex of a parabola y = ax^2 + bx + c is given by the formula x = -b / (2a). Here, a=1 and b=-4. So, x = -(-4) / (2*1) = 4 / 2 = 2.
47
If α and β are the roots of x^2 + 5x + 6 = 0, what is the value of αβ?
Answer:
6
The product of the roots (αβ) for a quadratic equation ax^2 + bx + c = 0 is defined as c/a. For x^2 + 5x + 6 = 0, c = 6 and a = 1. Therefore, αβ = 6/1 = 6.
48
If α and β are the roots of 2x^2 - 4x + 1 = 0, what is the value of 1/α + 1/β?
Answer:
4
The expression 1/α + 1/β simplifies to (α + β) / (αβ). From the equation, sum (α + β) = -(-4)/2 = 2. The product (αβ) = 1/2. Thus, the value is 2 / (1/2) = 4.
49
If α and β are the roots of x^2 - x - 1 = 0, find the value of α^2 + β^2.
Answer:
3
From the equation, sum of roots (α + β) = 1 and product (αβ) = -1. We use the identity α^2 + β^2 = (α + β)^2 - 2αβ. Substituting the values gives (1)^2 - 2(-1) = 1 + 2 = 3.
50
Which of the following is the correct quadratic formula?
Answer:
(-b ± sqrt(b^2 - 4ac)) / 2a
The quadratic formula, derived from completing the square of ax^2 + bx + c = 0, is x = (-b ± sqrt(b^2 - 4ac)) / 2a. It gives the roots of any generic quadratic equation.